SOLUTION: Prove that every positive integer a, written in the base 10, a^5 and a have the same last digit.
Algebra.Com
Question 520127: Prove that every positive integer a, written in the base 10, a^5 and a have the same last digit.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
The last digit of an integer is the remainder when the number is divided by 10. Any two natural numbers 
share the same last digit exactly when )
(Note: 
denotes "divides" and 
denotes "does not divide")
First prove )
: If 
is even, then 
is even and the difference of two even numbers is even. If is odd, then is odd, and the difference of two odd numbers is even. Thus .
Next prove )
. Fermat's Little Theorem: If 
is prime, for any integer 
, )
Hence 
for some integers 
. That means 
. Since 
, so 
since 2 is prime. Then 
where 
is an integer, and then 
.
Therefore )
John
My calculator said it, I believe it, that settles it
RELATED QUESTIONS
Prove that there is a positive integer that can be written as the sum of squares of... (answered by Edwin McCravy)
Can you please help to answer the following question:
.
When a positive integer N is (answered by khwang)
How many positive integers have the same number of digits when written in base 5 and in... (answered by greenestamps)
Determine all the positive integers that are written using 3 digits both in base 5 and in (answered by greenestamps)
Prove or disprove the following: The expression n2 – n +41 produces a prime number for... (answered by richard1234)
Find the 29th smallest positive integer that can be written only using the digits zero... (answered by LinnW)
Let n be a positive integer greater than 1. We call n prime if the only positive integers (answered by jim_thompson5910)
Find the biggest positive integral value that CANNOT be written as the sum of a positive... (answered by CubeyThePenguin,ikleyn)
Jane selects a positive two digit integer. She multiplies the ones digit by five and the... (answered by Edwin McCravy)
